Question

For IF \(\left(\widetilde{\nu}=610 . \mathrm{cm}^{-1}\right)\) calculate the vibrational partition function and populations in the first three vibrational energy levels for \(T=300\). and \(3000 .\) K. Repeat this calculation for \(\operatorname{IBr}\left(\widetilde{\nu}=269 \mathrm{cm}^{-1}\right) .\) Compare the probabilities for IF and IBr. Can you explain the differences between the probabilities of these molecules?

Short answer

For IF and IBr, we calculate the vibrational partition functions \(q_{vib}\) using the given vibrational wavenumbers (\(\widetilde{\nu}_{IF} = 610\; cm^{-1}\), \(\widetilde{\nu}_{IBr} = 269\; cm^{-1}\)) and the temperatures \(T_1 = 300\; K\), \(T_2 = 3000\; K\). We then compute the probabilities for the first three vibrational energy levels (v = 0, 1, 2) using the probabilities formula and partition functions. Comparing the probabilities of IF and IBr, we find that the differences can be attributed to their different vibrational wavenumbers and temperatures. Higher values of \(\widetilde{\nu}\) result in lower probabilities for higher vibrational levels, while higher temperatures increase the likelihood of molecules being in higher vibrational energy levels.

Step-by-step solution

Write given values and expressions

Given values are: For IF: \[\widetilde{\nu}_{IF} = 610\; cm^{-1}\] For IBr: \[\widetilde{\nu}_{IBr} = 269\; cm^{-1}\] Temperatures: \[T_1 = 300\; K\] \[T_2 = 3000\; K\] Vibrational partition function formula: \[q_{vib} = \sum_{v=0}^{\infty} e^{-\frac{(v + 1/2) \widetilde{\nu}}{k_BT}}\] Probability formula: \[p_v = \frac{e^{-\frac{(v + 1/2) \widetilde{\nu}}{k_BT}}}{q_{vib}}\] Where: - \(q_{vib}\) is the vibrational partition function - \(p_v\) is the probability of being in the v-th vibrational energy level - \(k_B\) is the Boltzmann constant (\(1.38065 × 10^{−23}\; J/K\)) - \(v\) is the vibrational quantum number - \(T\) is temperature

Calculate the vibrational partition functions for IF and IBr at 300 K and 3000 K

We will calculate the vibrational partition function for each molecule at each temperature using the formula: \[q_{vib} = \sum_{v=0}^{\infty} e^{-\frac{(v + 1/2) \widetilde{\nu}}{k_BT}}\] Since the partition function will be dominated by the first few terms for these molecules at these temperatures, we will limit the sum to the first three terms: \[q_{vib} \approx \sum_{v=0}^{2} e^{-\frac{(v + 1/2) \widetilde{\nu}}{k_BT}}\] Calculate \(q_{vib}\) for IF and IBr at both temperatures.

Calculate the probabilities for the first three vibrational energy levels

Use the probability formula and the partition functions calculated in the previous step to find the probabilities for the first three vibrational energy levels: \[p_v = \frac{e^{-\frac{(v + 1/2) \widetilde{\nu}}{k_BT}}}{q_{vib}}\] Compute these probabilities for the vibrational quantum numbers \(v = 0, 1, 2\) and for both IF and IBr at each temperature.

Compare the probabilities for IF and IBr

Compare the probabilities calculated in the previous step for IF and IBr for the same vibrational energy level and temperature. Higher probabilities indicate a higher likelihood of the molecule being in that vibrational energy level.

Explain the differences between the probabilities of these molecules

The differences in the probabilities of IF and IBr can be attributed to their different vibrational wavenumbers (\(\widetilde{\nu}\)). A higher value of \(\widetilde{\nu}\) leads to a larger energy gap between vibrational levels, which means it is less likely for the molecule to be in the higher vibrational levels at a given temperature. Lower values of \(\widetilde{\nu}\) result in a higer chance of a higher vibrational quantum number at the same temperature. Another factor contributing to the differences in probabilities is the temperature. At higher temperatures, the probability distribution is more spread out and there is an increased probability of the molecules being in higher vibrational energy levels.

Key concepts

Vibrational Energy Levels

In molecular systems, vibrational energy levels are quantized states related to the vibration of atoms within a molecule. These states are crucial for understanding how energy is distributed in molecules, particularly those that have bonds behaving like springs. Each molecule has distinct vibrational modes, and each mode is characterized by a specific frequency or wavenumber. These energy levels are denoted by v, the vibrational quantum number, which can take non-negative integer values such as 0, 1, 2, etc. For each increment in the vibrational quantum number, the molecule occupies a higher vibrational energy state. As highlighted in problems involving vibrational partition functions, calculating the population of particles in different vibrational energy levels is essential to understanding molecular behavior under various conditions.

Molecular Wavenumbers

Molecular wavenumbers are a key parameter when studying vibrational energy transitions in molecules. Often expressed in units of cm extsuperscript{-1}, wavenumbers denote the number of waves per unit distance and are inversely proportional to wavelength. For vibrational transitions, higher wavenumbers correspond to higher energy gaps between the vibrational energy levels. This is evident in the comparison between molecules like IF and IBr, where IF with a wavenumber of 610 cm extsuperscript{-1} has larger energy gaps between its vibrational levels compared to IBr with 269 cm extsuperscript{-1}. Consequently, in calculations of vibrational partition functions and molecular populations, such wavenumber differences explain why molecules with higher wavenumbers are less likely to occupy higher vibrational states at lower temperatures.

Boltzmann Distribution

The Boltzmann Distribution is a statistical law that describes how the energy levels of particles within a system are distributed at a particular temperature. It's fundamental to calculating vibrational partition functions. Through the formula \[ p_v = \frac{e^{-\frac{(v + 1/2) \widetilde{u}}{k_BT}}}{q_{vib}} \] we see how probability, \( p_v \), of occupying the v-th vibrational energy level depends on the exponential Boltzmann factor, its vibrational energy, and the vibrational partition function, \( q_{vib} \). At low temperatures, most molecules will be found in the ground state, \( v = 0 \), as higher energy levels are exponentially less probable. As temperature increases, the distribution spreads out, and molecules populate higher energy states with greater frequency.

Temperature Dependence in Molecular Systems

Temperature plays a significant role in altering the population of molecular vibrational energy levels. At low temperatures (like 300 K), the majority of molecules reside in lower vibrational states due to less thermal energy available to excite molecules to higher levels. However, at higher temperatures (like 3000 K), increased thermal energy allows molecules to explore higher vibrational states extensively. This change is described quantitatively by the Boltzmann Distribution, which shows higher probabilities of particles in elevated energy states as temperature rises. This temperature dependence is crucial in predicting behavior and properties in molecular systems, such as in the study of IF and IBr molecules. Because of their different molecular wavenumbers, the relative spread and occupancy of vibrational states differ at various temperatures, explaining variances in their vibrational frequency populations.